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I followed the link given by Catweasel. I do not know what the line "address bar: indicates client=firefox-b-1-ab " means.
It did not help me find out why "client=firefox-b" is all-of-a-sudden attached to all my searches. I have to remove the offending stuff manually to get reasonable search results.

Since there's no answer here and this was the first hit on Google for the problem:

Catweazel had not explained himself directly, but indirectly by pointing to a Mozilla Firefox support thread, explaining what happens. Cf. post above.
Seems no-one can be bothered to follow links posted as an answer.
Anyway.
Thanks for explaining another way of solving the same (non-)issue.

I really question that the suffixed string "client=firefox-b" is an issue at all.
It is not caused by the Google search plugin.
It is something done by Firefox itself, as far as I can find out.

Firefox 63.0.3 on Mint e.g.:
Depending on which search engine is used, Firefox will append no client string or "client=firefox-b".
If you perform the same search with different search engines available in the dropdown list of the search edit box to the right of the address bar (Google, Duckduckgo, Bing, etc), you will find out that Firefox appends different strings or none at all.

PaleMoon 28.2.1 on Mint e.g.:
If you perform exactly the same searches using exactly the same search engines, but in a different browser, e.g. Pale Moon here, then you will see that the browser will append its own name to searches as well. E.g. Pale Moon cannot be bothered to append its own name when using Google search. But it appends its own name, when using Duckduckgo (its default seach engine).

Firefox 63.0.3 on Ubuntu Mate e.g.:
The same Firefox on Ubuntu Mate appends "client=Ubuntu-browser", when searching via Google. On some other search engines it appends only "ubuntu". On some search engines it does not append anything.

The search engine, which consistently does not seem to be interested in receiving any client identification via the URL, seems to be StartPage.

I suspect that appending a client identification to the URL when executing a search depends on which browser has got a financial agreement with a search engine. No agreement, no client appended. - But this is only my assumption, I may be wrong.

Privacy Issue?
But why is this appended client identification an issue? Do you consider it a privacy issue?
Any website, which you visit, including search engines, will happily be given more details about your browser by your browser than just the name "firefox" or "palemoon" or "chrome".
Whoever wants to track you does not depend on your browser appending its brand in the URL.

Irrespective of whether making sure Firefox does not append "client=firefox-b", when searching through Google, makes too much sense, I have tried to find any hint whether it can be suppressed, but without any new result so far.

That "client=firefox" stuff has been bugging me (it gets stuck into URLs you use somewhere else), so I removed google as per above, exited FF normally, restarted it (google no longer shows up), attempting to add it back resulted in:

Now I've removed everything but google, so it's finally the default. I'll add the others and see what happens.

Sometimes I really enjoy the way an super-simple task, which could and probably should consist of something like "add 'google.com' to some text file, and put it at the top so it's the default", is made so complicated and convoluted that it no longer works.

Edit4: now I can't add any search engines via the official firefox menu->addon because it claims the names already exist even though the search engines don't show up anywhere.

Edit5: Just in case everyone was worried about the outcome of this battle, continual fiddling around got google back (FF menu->ff addon) as the default without the "client=firefox" nonsense. Now I get "&ie=utf-8&oe=utf-8" nonsense, which isn't as bad.

Please edit your original post title to include [SOLVED] if/when it is solved!Your data and OS are backed up....right?Mint 18.3 Xfce/fluxbox/pulse-lessXubuntu 17.10/fluxbox/pulse-less

I really question that the suffixed string "client=firefox-b" is an issue at all.
It is not caused by the Google search plugin.
It is something done by Firefox itself, as far as I can find out.

...

I suspect that appending a client identification to the URL when executing a search depends on which browser has got a financial agreement with a search engine. No agreement, no client appended. - But this is only my assumption, I may be wrong.

I obviously didn't explain the issue correctly.

I don't mean the URL in the Address bar.

I normally search by highlighting some text on a page and then use the Search option in the Context Menu.
When the result page/tab opens, the text I highlighted has "client=firefox-b" appended, which usually causes an incorrect search to be performed.
Manually deleting "client=firefox-b" from the text in the Results Page Search Box and searching again gives a different set of results.

Privacy Issue?
But why is this appended client identification an issue? Do you consider it a privacy issue?
Any website, which you visit, including search engines, will happily be given more details about your browser by your browser than just the name "firefox" or "palemoon" or "chrome".
Whoever wants to track you does not depend on your browser appending its brand in the URL.

I find it annoying NOT because it's a privacy issue (although it probably is) but because it means that I need to perform each search twice.

It seems that since I've complained about the issue FF + Google Search has been behaving itself (Murphy's Law).
The next time it starts misbehaving I'll try to get some screen shots.

That "client=firefox" stuff has been bugging me (it gets stuck into URLs you use somewhere else), so I removed google as per above, exited FF normally, restarted it (google no longer shows up), attempting to add it back resulted in: